The Internal Structure of Predicates and Names

This series of volumes is meant to extend the scope of what we can formalize in classical predicate logic, and in doing so see the real limitations of what can be done.

In the first section the standard of modern formal logic, classical predicate logic with equality, is set out, drawing on the full development in An Introduction to Formal Logic.

In the second section classical predicate logic is extended to formalizereasoning that involves adverbs and relative adjectives by viewing those as modifiers of simpler predicates. What is normally taken to be atomic predicates, such as “barking loudly”, can then have internal structure. Reasoning that involves conjunctions of terms, as in “Tom and Dick lifted the table”, conjunctions of modifiers, conjunctions of predicates, and disjunctions of predicates can also be formalized by viewing them as part of the internal structure of atomic predicates.

The internal structure of names is the topic of the third and last section. Names for functions are used in classical predicate logic to form complex names are presented first. In our ordinary reasoning we use descriptions to form functions, such as “the wife of”, and descriptions to form names, such as “the cat that scratched Zoe”. To reason with those we can take account of their internal structure by dropping the assumption that every name must refer to a specific thing.

The formal systems that are developed here are not just formalisms but are meant to help us understand how to reason well. Many worked examples show how to use them. They also uncover limitations of the formal work. The analyses in the examples are tentative, presented with the hope of stimulating you to deeper and clearer analyses.

The work here proceeds by abstracting and creating formal models to formalize reasoning. By paying attention to the process of abstracting we gain insight into why we consider some reasoning to be good and some reasoning bad, and insight also into the deeper assumptions we make about the world on which our judgments rely. Questions about the metaphysics we assume for modern formal logic and the nature of formalizing have to be faced, most particularly the assumption that the world is made up of objects that we can name.

This work extends the scope of classical predicate logic by showing how to formalize reasoning that involves adverbs, relative adjectives, conjunctions of terms, conjunctions of modifiers, and conjunctions of predicates as part of the internal structure of atomic predicates. Descriptive names functions and non-referring names are also analyzed.

INTRODUCTION

BACKGROUND

1 Formal Logic

2 Classical Propositional Logic

3 Formal Theories of Reasoning Well and Limitations of Propositional Logic